Data_Inspection

knitr::opts_chunk$set(fig.width = 8,fig.height = 6)
library(tidyverse)

## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
## ✔ ggplot2 3.3.6      ✔ purrr   0.3.5 
## ✔ tibble  3.1.8      ✔ dplyr   1.0.10
## ✔ tidyr   1.2.1      ✔ stringr 1.4.1 
## ✔ readr   2.1.3      ✔ forcats 0.5.2 
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()

library(rayshader)
library(patchwork)
library(skimr)
library(visdat)
library(ggplot2)
library(GGally)

## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2

library(corrplot)

## corrplot 0.92 loaded

A first inspection of the dataset with 78 variables

importing the data

D = read.table("/Users/macbook/Documents/Bayesian Statistics/Project/Raw_data/LIPIDI/78 variabili/101_lipidi-PreProcessed-IM-Step1-Step2-Step4-Step5-101.txt")
sum(is.na(D))

## [1] 634087

the numbers of na is substantial

 vis_miss(D,warn_large_data = FALSE)

## Warning: `gather_()` was deprecated in tidyr 1.2.0.
## ℹ Please use `gather()` instead.
## ℹ The deprecated feature was likely used in the visdat package.
##   Please report the issue at <]8;;https://github.com/ropensci/visdat/issueshttps://github.com/ropensci/visdat/issues]8;;>.

the missing data is about 45%

skim(D)

Data summary

Name	D
Number of rows	18229
Number of columns	78
_______________________
Column type frequency:
numeric	78
________________________
Group variables	None

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
X401.18155749152	1561	0.91	0.14	0.07	0.02	0.09	0.13	0.18	0.81	▇▂▁▁▁
X402.175372221284	4352	0.76	0.09	0.04	0.02	0.06	0.08	0.11	0.38	▇▅▁▁▁
X409.217742652953	4174	0.77	0.10	0.11	0.03	0.06	0.07	0.10	1.34	▇▁▁▁▁
X419.246407498537	12919	0.29	0.06	0.01	0.02	0.05	0.05	0.06	0.14	▃▇▂▁▁
X426.123469726227	4124	0.77	0.13	0.07	0.03	0.09	0.12	0.15	1.25	▇▁▁▁▁
X437.311179462526	9143	0.50	0.13	0.05	0.04	0.10	0.12	0.15	0.52	▇▃▁▁▁
X442.117407247144	8813	0.52	0.22	0.13	0.03	0.14	0.18	0.26	1.37	▇▂▁▁▁
X447.212994469759	542	0.97	0.62	0.24	0.01	0.45	0.60	0.77	2.14	▃▇▂▁▁
X448.206629110332	11103	0.39	0.11	0.03	0.03	0.08	0.10	0.12	0.38	▇▇▁▁▁
X449.181170853303	11116	0.39	0.11	0.04	0.02	0.08	0.10	0.13	0.49	▇▅▁▁▁
X465.221110403551	9453	0.48	0.05	0.02	0.02	0.04	0.04	0.05	0.25	▇▁▁▁▁
X478.277226797819	4972	0.73	0.06	0.02	0.01	0.05	0.06	0.07	0.15	▁▇▅▁▁
X479.248654523272	12046	0.34	0.04	0.02	0.02	0.03	0.04	0.05	0.32	▇▁▁▁▁
X497.206369201144	10686	0.41	0.07	0.22	0.02	0.05	0.06	0.07	9.15	▇▁▁▁▁
X506.28595502602	13991	0.23	0.05	0.01	0.01	0.04	0.05	0.06	0.14	▂▇▂▁▁
X511.153597066032	11218	0.38	0.52	1.12	0.01	0.04	0.05	0.24	11.68	▇▁▁▁▁
X513.237647290215	14480	0.21	0.08	0.06	0.01	0.04	0.06	0.09	0.77	▇▁▁▁▁
X524.279211549971	4709	0.74	0.10	0.03	0.02	0.08	0.09	0.12	0.30	▂▇▂▁▁
X525.261547813485	12627	0.31	0.05	0.01	0.01	0.04	0.05	0.06	0.12	▁▇▃▁▁
X538.265072270527	9789	0.46	0.05	0.01	0.02	0.04	0.05	0.05	0.10	▂▇▅▁▁
X552.2715207686	10676	0.41	0.06	0.01	0.01	0.05	0.06	0.06	0.12	▁▇▇▁▁
X553.239346145671	2177	0.88	0.19	0.09	0.01	0.13	0.18	0.24	0.69	▅▇▂▁▁
X558.153387011278	14329	0.21	0.06	0.02	0.02	0.05	0.05	0.07	0.26	▇▂▁▁▁
X566.28121499168	14220	0.22	0.08	0.02	0.02	0.06	0.08	0.09	0.17	▁▇▆▁▁
X568.285314369757	9344	0.49	0.16	0.05	0.02	0.12	0.15	0.19	0.42	▂▇▃▁▁
X576.257761688406	10669	0.41	0.06	0.02	0.02	0.04	0.05	0.06	0.23	▇▅▁▁▁
X577.277943537651	9942	0.45	0.12	0.05	0.03	0.09	0.11	0.14	0.52	▇▅▁▁▁
X578.280306512124	10705	0.41	0.07	0.03	0.02	0.06	0.07	0.09	0.29	▇▅▁▁▁
X592.267287982367	12440	0.32	0.08	0.03	0.02	0.07	0.08	0.10	0.26	▆▇▁▁▁
X595.24167872285	13615	0.25	0.08	0.02	0.03	0.07	0.08	0.09	0.16	▁▇▆▁▁
X599.312316651348	1947	0.89	0.15	0.06	0.02	0.11	0.14	0.17	0.92	▇▁▁▁▁
X600.287990847865	9059	0.50	0.08	0.02	0.02	0.06	0.08	0.09	0.31	▇▇▁▁▁
X614.280680723305	11990	0.34	0.06	0.02	0.02	0.05	0.06	0.07	0.22	▇▆▁▁▁
X616.374126264944	13264	0.27	0.05	0.01	0.02	0.04	0.05	0.05	0.09	▁▇▇▂▁
X652.361088301576	9561	0.48	0.08	0.03	0.02	0.06	0.08	0.10	0.21	▃▇▅▁▁
X653.324143837042	2179	0.88	0.24	0.10	0.01	0.18	0.23	0.29	1.28	▇▃▁▁▁
X654.321418202011	10592	0.42	0.14	0.04	0.02	0.11	0.14	0.17	0.35	▁▇▅▁▁
X666.348091732679	10727	0.41	0.12	0.04	0.02	0.09	0.11	0.15	0.31	▂▇▅▁▁
X668.350119384666	8434	0.54	0.09	0.04	0.02	0.06	0.08	0.11	0.26	▆▇▃▁▁
X697.336385172531	765	0.96	0.28	0.10	0.02	0.22	0.28	0.35	0.63	▂▆▇▂▁
X698.342468583488	6941	0.62	0.18	0.06	0.01	0.14	0.18	0.22	0.45	▂▇▇▁▁
X713.313334386064	7293	0.60	0.21	0.09	0.02	0.16	0.21	0.25	1.06	▇▅▁▁▁
X723.369767544851	9485	0.48	0.08	0.02	0.01	0.07	0.08	0.10	0.16	▁▅▇▃▁
X724.357765223416	11747	0.36	0.06	0.02	0.01	0.04	0.06	0.07	0.17	▂▇▂▁▁
X726.353588041765	10899	0.40	0.15	0.04	0.02	0.13	0.15	0.17	0.48	▁▇▁▁▁
X737.153366400938	4088	0.78	0.14	0.07	0.01	0.09	0.13	0.17	0.73	▇▃▁▁▁
X753.367749586126	8284	0.55	0.13	0.04	0.02	0.10	0.13	0.15	0.32	▂▇▆▁▁
X762.346785523875	12992	0.29	0.08	0.02	0.02	0.06	0.08	0.09	0.20	▂▇▃▁▁
X764.372631793774	11446	0.37	0.05	0.02	0.01	0.04	0.05	0.07	0.14	▃▇▃▁▁
X774.982938560717	8359	0.54	0.19	0.06	0.02	0.15	0.19	0.23	0.53	▁▇▃▁▁
X775.976648338383	13999	0.23	0.08	0.02	0.02	0.06	0.08	0.09	0.17	▂▇▇▂▁
X778.403735371109	9983	0.45	1.18	0.53	0.02	0.82	1.14	1.52	4.02	▃▇▂▁▁
X779.39508853726	5855	0.68	0.50	0.26	0.01	0.29	0.46	0.68	1.83	▇▇▃▁▁
X780.399514391103	8392	0.54	0.27	0.13	0.02	0.17	0.24	0.35	0.97	▇▇▃▁▁
X793.327494359355	14448	0.21	0.08	0.03	0.01	0.06	0.08	0.10	0.17	▂▆▇▅▁
X806.438265275953	1282	0.93	1.50	1.55	0.02	0.37	0.99	2.04	10.18	▇▂▁▁▁
X807.430315603065	2270	0.88	0.93	0.81	0.01	0.27	0.70	1.37	4.61	▇▃▂▁▁
X808.418052996172	4205	0.77	0.48	0.34	0.01	0.21	0.40	0.70	2.01	▇▅▂▁▁
X822.416323513071	2717	0.85	1.34	0.84	0.02	0.64	1.25	1.92	5.52	▇▇▃▁▁
X823.418850404014	3984	0.78	0.87	0.51	0.02	0.48	0.79	1.23	2.88	▇▇▅▁▁
X836.412896788889	9507	0.48	0.22	0.09	0.01	0.15	0.20	0.28	0.62	▂▇▃▁▁
X862.437600249503	280	0.98	2.79	2.42	0.01	0.66	2.24	4.31	13.04	▇▃▂▁▁
X863.420550439	1936	0.89	1.53	1.15	0.01	0.54	1.31	2.31	6.36	▇▅▃▁▁
X888.444482031706	1435	0.92	9.36	6.70	0.02	3.84	8.39	13.40	40.59	▇▆▂▁▁
X889.43378336125	4013	0.78	5.00	3.82	0.03	2.09	4.21	7.08	22.78	▇▅▂▁▁
X890.455559065043	1940	0.89	11.17	8.12	0.03	4.38	9.83	16.57	44.94	▇▆▃▁▁
X904.460038847805	10652	0.42	2.05	1.79	0.03	0.37	1.77	3.24	9.68	▇▅▂▁▁
X906.471649609416	673	0.96	12.42	9.56	0.02	4.10	11.08	18.50	51.34	▇▆▃▁▁
X907.47494585213	8186	0.55	8.65	5.02	0.02	4.82	8.24	12.22	26.62	▆▇▆▂▁
X908.485206763087	3471	0.81	3.38	2.22	0.01	1.68	3.06	4.85	12.40	▇▇▃▁▁
X933.449659746903	11508	0.37	0.10	0.03	0.01	0.07	0.10	0.12	0.24	▂▇▆▁▁
X934.487399616051	13583	0.25	0.28	0.12	0.01	0.20	0.27	0.36	0.77	▂▇▅▁▁
X942.439351429176	13376	0.27	0.08	0.03	0.01	0.05	0.08	0.10	0.24	▅▇▃▁▁
X943.432502731421	7473	0.59	0.10	0.04	0.01	0.07	0.09	0.12	0.32	▃▇▂▁▁
X944.447551703558	10962	0.40	0.07	0.03	0.01	0.05	0.06	0.08	0.29	▇▅▁▁▁
X965.462816974066	2475	0.86	0.10	0.04	0.01	0.07	0.09	0.12	0.33	▅▇▂▁▁
X966.464009229031	6153	0.66	0.06	0.02	0.01	0.04	0.05	0.07	0.17	▅▇▂▁▁
X967.456646158017	9342	0.49	0.04	0.01	0.01	0.03	0.04	0.05	0.10	▂▇▃▁▁

we observe that the missing data is not uniform in the mz, there are some values for which only 20 - 30% of the pixel have a value, and this tends to be small

we replace the missing data with 0 since it means the data for that mz was under threshold

D0 = D
D0[is.na(D0)] = 0

preliminary plotts

look at the distribution and correlation of the data, beginning with similar mz probably ggpairs is not the most efficient, did not change the labels

ggpairs(D,columns = 1:7)

ggpairs(D,columns = 8:14)

ggpairs(D,columns = 15:21)

ggpairs(D,columns = 16:28)

ggpairs(D,columns = 28:35)

ggpairs(D,columns = 36:42)

ggpairs(D,columns = 43:49)

ggpairs(D,columns = 50:56)

we see a lot of correlation in the data especially below

ggpairs(D,columns = 57:63)

ggpairs(D,columns = 64:70)

ggpairs(D,columns = 70:78)

• do the zeros added contribute a lot to the correlation?

• the peaks that we have around 800 mz are all correlated and are basically the same information

• this high correlation explains the performance of pca

correlation matrix

cm <- cor(D0)
corrplot(cm, method = "color")

the blue blob is the observation from above

pixels = read.table("/Users/macbook/Documents/Bayesian Statistics/Project/Raw_data/LIPIDI/78 variabili/101_lipidi-PreProcessed-XYCoordinates-Step1-Step2-Step4-Step5-101.txt")
colnames(D0) = substr(colnames(D0),1,4)
colnames(pixels) = c("x","y")

Create the datasets we will need:

• Data_long: contains the 18k pixels on the rows. The first two columns are the coordinates, the remaining 78 are the values recorded for different m.z.

• Data_array: contains a cube with the 78 slices on the third dimension. Not all the pixels are recored in each rectangle: beware of NA’s.

Data_long            = as_tibble(data.frame( pixels, D0 ))
max_number_of_pixels = apply(Data_long[,1:2],2,max)

Data_array = matrix(NA,max_number_of_pixels[1],max_number_of_pixels[2])

Data_array = array(NA,c(max_number_of_pixels[1],max_number_of_pixels[2],ncol(D0)))

sum(is.na(D0))

## [1] 0

# there must be a better way to do this, but it's sunday morning, please be patient...
for(k in 1:ncol(D0)){
  for(i in 1:nrow(Data_long)){
  Data_array[Data_long$x[i],Data_long$y[i],k] = D0[i,k]
  }
}

dim(Data_array)

##   x   y     
## 157 178  78

Data_very_long = reshape2::melt(Data_long,c("x","y")) %>% mutate(pixel_ind = paste0(x,"_",y), value_ind = rep(1:nrow(Data_long),ncol(D0)))
Data_very_long = Data_very_long %>% group_by(pixel_ind) %>% mutate(n = row_number()) %>% ungroup() %>% mutate(mz = as.numeric(substr(variable,2,4)))

Data_very_long = reshape2::melt(Data_long,c("x","y")) %>% mutate(pixel_ind = paste0(x,"_",y), value_ind = rep(1:nrow(Data_long),ncol(D0)))

Data_very_long = Data_very_long %>% group_by(pixel_ind) %>% mutate(n = row_number()) %>% ungroup() %>% mutate(mz = as.numeric(substr(variable,2,4)))


# subsampling to get a faster plot and not drain memory
sub_ind = sample(unique(Data_very_long$pixel_ind),1000)
# just to get the gist:
ggplot(Data_very_long %>% filter(pixel_ind %in% sub_ind))+
  geom_path(aes(x = mz, y = value, 
                col=pixel_ind, 
                group = pixel_ind),alpha=.5)+theme_bw()+theme(legend.position = "none")+xlab("m.z")+scale_color_viridis_d(option = "A")+
  scale_x_continuous(n.breaks = 20)

investigating the different peaks

the first mz don’t contain a lot of info, quite noisy

we investigate the spike arround 500

the spike is in mz 511 on the edge
- possible problem of the instrument on the edge of the brain - outlier ? the rest is just noise

spike arround 775

the spike is relative to 778 779 780

spike arround 800 and 825

very similar to each other

after 850

this are all the same, it is the high correlated blob in the correlation matrix

there is another peak around 900

the peak is in 906 907 908 which are very similar

the rest is not that itresting, very low values resamble kind on veramble the previuus structure but basically noise

A comprehensive look at the peaks

exept for the 511 which is in the edge, the peaks are all correlarend and therefore are all the same information

PCA

pca = princomp(D0)
plot(pca)

summary(pca)

## Importance of components:
##                           Comp.1     Comp.2    Comp.3      Comp.4     Comp.5
## Standard deviation     16.063465 3.74669709 1.8050014 1.557047404 1.16570821
## Proportion of Variance  0.917156 0.04989556 0.0115803 0.008617244 0.00482997
## Cumulative Proportion   0.917156 0.96705159 0.9786319 0.987249138 0.99207911
##                             Comp.6      Comp.7       Comp.8       Comp.9
## Standard deviation     0.775473863 0.732621491 0.5247242544 0.4176023644
## Proportion of Variance 0.002137465 0.001907761 0.0009786491 0.0006198558
## Cumulative Proportion  0.994216573 0.996124335 0.9971029837 0.9977228394
##                             Comp.10      Comp.11      Comp.12      Comp.13
## Standard deviation     0.3441918928 0.3285733456 0.2614626699 0.2329424828
## Proportion of Variance 0.0004210814 0.0003837332 0.0002429876 0.0001928689
## Cumulative Proportion  0.9981439208 0.9985276540 0.9987706416 0.9989635105
##                             Comp.14      Comp.15      Comp.16      Comp.17
## Standard deviation     0.1901943823 0.1719970735 1.542005e-01 1.494558e-01
## Proportion of Variance 0.0001285761 0.0001051494 8.451544e-05 7.939449e-05
## Cumulative Proportion  0.9990920865 0.9991972360 9.992818e-01 9.993611e-01
##                             Comp.18      Comp.19      Comp.20      Comp.21
## Standard deviation     1.437853e-01 1.277647e-01 1.156808e-01 1.124762e-01
## Proportion of Variance 7.348417e-05 5.802117e-05 4.756501e-05 4.496621e-05
## Cumulative Proportion  9.994346e-01 9.994927e-01 9.995402e-01 9.995852e-01
##                             Comp.22      Comp.23      Comp.24      Comp.25
## Standard deviation     1.112338e-01 1.101552e-01 9.828352e-02 8.492948e-02
## Proportion of Variance 4.397825e-05 4.312951e-05 3.433414e-05 2.563786e-05
## Cumulative Proportion  9.996292e-01 9.996723e-01 9.997066e-01 9.997323e-01
##                             Comp.26      Comp.27      Comp.28      Comp.29
## Standard deviation     8.148553e-02 7.341267e-02 7.132058e-02 0.0693419808
## Proportion of Variance 2.360075e-05 1.915609e-05 1.807984e-05 0.0000170906
## Cumulative Proportion  9.997559e-01 9.997750e-01 9.997931e-01 0.9998101895
##                             Comp.30      Comp.31      Comp.32      Comp.33
## Standard deviation     6.664939e-02 6.041977e-02 0.0591444918 5.537155e-02
## Proportion of Variance 1.578909e-05 1.297546e-05 0.0000124335 1.089778e-05
## Cumulative Proportion  9.998260e-01 9.998390e-01 0.9998513875 9.998623e-01
##                             Comp.34      Comp.35      Comp.36      Comp.37
## Standard deviation     5.324759e-02 5.195975e-02 0.0500307473 4.844369e-02
## Proportion of Variance 1.007778e-05 9.596191e-06 0.0000088969 8.341404e-06
## Cumulative Proportion  9.998724e-01 9.998820e-01 0.9998908562 9.998992e-01
##                             Comp.38      Comp.39      Comp.40      Comp.41
## Standard deviation     4.698990e-02 4.254707e-02 3.900727e-02 3.613078e-02
## Proportion of Variance 7.848266e-06 6.434341e-06 5.408237e-06 4.640016e-06
## Cumulative Proportion  9.999070e-01 9.999135e-01 9.999189e-01 9.999235e-01
##                             Comp.42      Comp.43      Comp.44      Comp.45
## Standard deviation     3.413272e-02 3.298352e-02 3.136910e-02 3.051343e-02
## Proportion of Variance 4.141012e-06 3.866862e-06 3.497589e-06 3.309381e-06
## Cumulative Proportion  9.999277e-01 9.999315e-01 9.999350e-01 9.999383e-01
##                             Comp.46      Comp.47      Comp.48      Comp.49
## Standard deviation     2.993919e-02 2.931011e-02 2.866060e-02 2.810079e-02
## Proportion of Variance 3.185993e-06 3.053513e-06 2.919680e-06 2.806738e-06
## Cumulative Proportion  9.999415e-01 9.999446e-01 9.999475e-01 9.999503e-01
##                             Comp.50      Comp.51      Comp.52      Comp.53
## Standard deviation     2.706671e-02 2.659691e-02 2.635536e-02 2.555264e-02
## Proportion of Variance 2.603969e-06 2.514359e-06 2.468896e-06 2.320792e-06
## Cumulative Proportion  9.999529e-01 9.999554e-01 9.999579e-01 9.999602e-01
##                             Comp.54      Comp.55      Comp.56      Comp.57
## Standard deviation     2.524454e-02 2.487671e-02 2.455916e-02 2.438560e-02
## Proportion of Variance 2.265165e-06 2.199636e-06 2.143836e-06 2.113642e-06
## Cumulative Proportion  9.999625e-01 9.999647e-01 9.999668e-01 9.999689e-01
##                             Comp.58      Comp.59      Comp.60      Comp.61
## Standard deviation     2.377389e-02 2.311280e-02 2.300476e-02 2.231907e-02
## Proportion of Variance 2.008932e-06 1.898759e-06 1.881049e-06 1.770585e-06
## Cumulative Proportion  9.999709e-01 9.999728e-01 9.999747e-01 9.999765e-01
##                             Comp.62      Comp.63      Comp.64      Comp.65
## Standard deviation     2.215556e-02 2.197574e-02 2.183727e-02 2.111930e-02
## Proportion of Variance 1.744738e-06 1.716532e-06 1.694967e-06 1.585346e-06
## Cumulative Proportion  9.999782e-01 9.999800e-01 9.999817e-01 9.999832e-01
##                             Comp.66      Comp.67      Comp.68      Comp.69
## Standard deviation     2.065927e-02 2.060456e-02 2.037009e-02 1.987790e-02
## Proportion of Variance 1.517033e-06 1.509007e-06 1.474860e-06 1.404448e-06
## Cumulative Proportion  9.999848e-01 9.999863e-01 9.999877e-01 9.999891e-01
##                             Comp.70      Comp.71      Comp.72      Comp.73
## Standard deviation     1.939794e-02 1.925609e-02 1.916565e-02 1.876678e-02
## Proportion of Variance 1.337445e-06 1.317955e-06 1.305605e-06 1.251827e-06
## Cumulative Proportion  9.999905e-01 9.999918e-01 9.999931e-01 9.999944e-01
##                             Comp.74      Comp.75      Comp.76      Comp.77
## Standard deviation     1.863689e-02 1.821375e-02 1.753698e-02 1.738131e-02
## Proportion of Variance 1.234558e-06 1.179135e-06 1.093136e-06 1.073816e-06
## Cumulative Proportion  9.999956e-01 9.999968e-01 9.999979e-01 9.999989e-01
##                             Comp.78
## Standard deviation     1.727547e-02
## Proportion of Variance 1.060778e-06
## Cumulative Proportion  1.000000e+00

the pca works well because of the correlation

PCA1 = ggplot(Data_long %>% mutate(pca1 = pca$scores[,1]))+ theme_bw()+
  geom_tile(aes(x=x,y=y,fill = pca1))+scale_fill_viridis_c(option = "A",na.value = "red")+
  theme_void()+theme(legend.position = "bottom")
PCA2 = ggplot(Data_long %>% mutate(pca2 = pca$scores[,2]))+ theme_bw()+
  geom_tile(aes(x=x,y=y,fill = pca2))+scale_fill_viridis_c(option = "A",na.value = "red")+
  theme_void()+theme(legend.position = "bottom")

PCA1+PCA2

Note: we are always using the same info

if we invert the colormap

PCA1v2 = ggplot(Data_long %>% mutate(pca1 = pca$scores[,1]))+ theme_bw()+
  geom_tile(aes(x=x,y=y,fill = -1*pca1))+scale_fill_viridis_c(option = "A",na.value = "red")+
  theme_void()+theme(legend.position = "bottom")

PCA1v2 + P6

the data is clearly dominated by this peaks that are highly correlated and spatially correlated as well

Plotting the principal components

first component

second component

do we need to be smarter and explore the non dominant factors or is enough?

sessionInfo()

## R version 4.2.1 (2022-06-23)
## Platform: x86_64-apple-darwin17.0 (64-bit)
## Running under: macOS Big Sur ... 10.16
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRlapack.dylib
## 
## locale:
## [1] it_IT.UTF-8/it_IT.UTF-8/it_IT.UTF-8/C/it_IT.UTF-8/it_IT.UTF-8
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] corrplot_0.92     GGally_2.1.2      visdat_0.5.3      skimr_2.1.4      
##  [5] patchwork_1.1.2   rayshader_0.24.10 forcats_0.5.2     stringr_1.4.1    
##  [9] dplyr_1.0.10      purrr_0.3.5       readr_2.1.3       tidyr_1.2.1      
## [13] tibble_3.1.8      ggplot2_3.3.6     tidyverse_1.3.2  
## 
## loaded via a namespace (and not attached):
##  [1] httr_1.4.4          sass_0.4.2          viridisLite_0.4.1  
##  [4] jsonlite_1.8.2      foreach_1.5.2       modelr_0.1.9       
##  [7] bslib_0.4.0         assertthat_0.2.1    highr_0.9          
## [10] googlesheets4_1.0.1 cellranger_1.1.0    yaml_2.3.6         
## [13] progress_1.2.2      pillar_1.8.1        backports_1.4.1    
## [16] glue_1.6.2          digest_0.6.30       RColorBrewer_1.1-3 
## [19] rvest_1.0.3         colorspace_2.0-3    plyr_1.8.7         
## [22] htmltools_0.5.3     pkgconfig_2.0.3     broom_1.0.1        
## [25] haven_2.5.1         scales_1.2.1        tzdb_0.3.0         
## [28] googledrive_2.0.0   farver_2.1.1        generics_0.1.3     
## [31] ellipsis_0.3.2      cachem_1.0.6        withr_2.5.0        
## [34] repr_1.1.4          cli_3.4.1           magrittr_2.0.3     
## [37] crayon_1.5.2        readxl_1.4.1        evaluate_0.17      
## [40] fs_1.5.2            fansi_1.0.3         doParallel_1.0.17  
## [43] xml2_1.3.3          tools_4.2.1         prettyunits_1.1.1  
## [46] hms_1.1.2           gargle_1.2.1        lifecycle_1.0.3    
## [49] munsell_0.5.0       reprex_2.0.2        compiler_4.2.1     
## [52] jquerylib_0.1.4     rlang_1.0.6         grid_4.2.1         
## [55] iterators_1.0.14    rstudioapi_0.14     htmlwidgets_1.5.4  
## [58] labeling_0.4.2      base64enc_0.1-3     rmarkdown_2.17     
## [61] gtable_0.3.1        codetools_0.2-18    reshape_0.8.9      
## [64] DBI_1.1.3           reshape2_1.4.4      R6_2.5.1           
## [67] lubridate_1.8.0     knitr_1.40          fastmap_1.1.0      
## [70] utf8_1.2.2          stringi_1.7.8       parallel_4.2.1     
## [73] Rcpp_1.0.9          vctrs_0.4.2         rgl_0.110.2        
## [76] dbplyr_2.2.1        tidyselect_1.2.0    xfun_0.33